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To solve the problem of finding the greatest of three consecutive even integers whose sum is 12, let's break it down step by step:

### Step 1: Define the three consecutive even integers.
Let the first even integer be \( x \). The next two consecutive even integers will be \( x + 2 \) and \( x + 4 \).

### Step 2: Write the equation.
Since the sum of these integers is given as 12, we can write:

\[
x + (x + 2) + (x + 4) = 12
\]

### Step 3: Simplify the equation.
\[
x + x + 2 + x + 4 = 12
\]
\[
3x + 6 = 12
\]

### Step 4: Solve for \( x \).
Subtract 6 from both sides:
\[
3x = 12 - 6
\]
\[
3x = 6
\]
Now divide by 3:
\[
x = 2
\]

### Step 5: Find the consecutive integers.
- The first integer is \( x = 2 \).
- The second integer is \( x + 2 = 4 \).
- The third integer is \( x + 4 = 6 \).

### Step 6: Identify the greatest integer.
The greatest of these integers is **6**.

Would you like further clarification or more details?

Here are five related questions to extend the discussion:
1. What if the sum of three consecutive odd integers was 15? What would the greatest integer be?
2. Can you find the sum of four consecutive even integers adding up to 24?
3. How would the equation change if the three integers were consecutive odd numbers?
4. How can we generalize a formula to find consecutive integers if the sum is given?
5. What are consecutive integers and how do they differ from consecutive even or odd integers?

**Tip**: When dealing with consecutive numbers (even, odd, or just integers), always express them in terms of a single variable for ease of solving the equation.

Three consecutive even integers add up to 12. What is the greatest of these integers?

Learn how to find the greatest of three consecutive even integers whose sum is 12. This problem demonstrates the use of algebra and linear equations, suitable for students in Grades 6-8. Step-by-step solution included.

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